I am asked to find the first three terms in the taylor series of the function
$$ f(x)=(x-1)\ln x $$
around $x_0=0$.
Then to find the maximum error in my approximation in the interval $[0.5,1.5]$.
My question is how could I find the first three terms of the function since the first term is $f(x_0)=f(0)=\infty$
Hint:
From the well-known $$\ln(1+t)=t-\frac{t^2}2+\frac{t^3}3-\cdots,$$ you get that
$$(x-1)\ln(x)=(x-1)^2-\frac{(x-1)^3}2+\frac{(x-1)^4}3-\cdots$$